Preamble

A Weighted Compact Nonlinear Scheme with Adaptive-Order Interpolation (WCNS-AO) for Euler Equations

Authors: Hu Yinggang, Jiang Yanqun, Huang Xiaoqian
(School of Mathematics and Physics, Southwest University of Science and Technology)

Abstract

The Euler equations represent a typical class of nonlinear hyperbolic conservation laws, the solutions of which may contain discontinuities. This characteristic often leads to non-physical oscillations near discontinuities when high-order numerical schemes are employed. In this study, a new weighted compact nonlinear scheme (WCNS-AO) for the Euler equations is developed based on the concept of adaptive-order (AO) interpolation. Building upon the classical five-point stencil, this scheme utilizes a nonlinear weighted interpolation technique with adaptive order, which blends low-order interpolations on sub-stencils with high-order interpolation on the global stencil to improve the performance of the original framework. In the WCNS-AO scheme, the spatial derivatives of the flux at cell nodes are calculated using a hybrid six-order central difference scheme involving both nodes and half-nodes. The fluxes at cell interfaces are computed using the adaptive-order nonlinear interpolation method in characteristic form. A high-order TVD Runge-Kutta method is employed to solve the semi-discrete equations obtained after spatial discretization of the Euler equations. Numerical results demonstrate that the WCNS-AO scheme achieves its fifth-order design accuracy in smooth regions. Furthermore, compared to the classical fifth-order WCNS-Z and WENO-AO schemes, the proposed method exhibits superior resolution and shock-capturing capabilities.

1. Introduction

The Euler equations are a fundamental set of nonlinear hyperbolic conservation laws used to describe inviscid fluid flow. A primary challenge in solving these equations numerically is the presence of discontinuities, such as shock waves and contact surfaces. High-order numerical methods are essential for capturing fine-scale structures in complex flows; however, traditional high-order linear schemes often suffer from Gibbs oscillations near discontinuities. To address this, nonlinear schemes such as the Weighted Essentially Non-Oscillatory (WENO) and Weighted Compact Nonlinear Schemes (WCNS) have been widely developed.

WCNS, in particular, combines the high-order spectral-like resolution of compact schemes with the robust shock-capturing capabilities of WENO-type interpolations. By separating the interpolation process from the derivative calculation, WCNS provides a flexible framework for high-fidelity simulations. This paper introduces an enhancement to this framework by incorporating adaptive-order (AO) interpolation logic.

2. Numerical Method

2.1 Spatial Discretization

The governing Euler equations in one dimension can be expressed as:
$$\frac{\partial \mathbf{U}}{\partial t} + \frac{\partial \mathbf{F}(\mathbf{U})}{\partial x} = 0$$
where $\mathbf{U}$ is the vector of conserved variables and $\mathbf{F}(\mathbf{U})$ is the flux vector. To discretize the spatial derivative at a grid point $x_i$, we employ a hybrid six-order central difference scheme that utilizes values at both nodes and half-nodes ($x_{i \pm 1/2}$).

2.2 Adaptive-Order Interpolation (WCNS-AO)

The core of the WCNS-AO scheme lies in the reconstruction of interface values. Unlike standard WCNS, which uses a fixed combination of candidate stencils, the AO approach dynamically adjusts the interpolation order based on the local smoothness of the solution.

On a five-point global stencil, we consider several sub-stencils. The adaptive-order technique computes a set of non-negative weights $\omega_k$ based on smoothness indicators $\beta_k$. The final interpolated value at the interface $x_{i+1/2}$ is defined as a weighted combination:
$$\hat{f}{i+1/2} = \omega} \mathcal{P{hi} + \omega} \mathcal{P{lo}$$
where $\mathcal{P}$ represents the lower-order reconstructions on smaller sub-stencils. This formulation ensures that the scheme maintains fifth-order accuracy in smooth regions by recovering the central high-order stencil, while automatically reverting to lower-order, more dissipative stencils near shocks to suppress oscillations.}$ represents the high-order polynomial reconstruction on the global stencil and $\mathcal{P}_{lo

2.3 Characteristic Decomposition and Time Integration

To improve stability and reduce oscillations for system equations, the interpolation is performed in characteristic space. The physical variables are projected onto the eigenvectors of the Jacobian matrix $\mathbf{A} = \partial \mathbf{F} / \partial \mathbf{U}$. After the nonlinear interpolation is completed in the characteristic fields, the values are transformed back to physical space to compute the numerical fluxes.

For the temporal evolution, we utilize a high-order Total Variation Diminishing (TVD) Runge-Kutta method to solve the resulting semi-discrete system:
$$\frac{d\mathbf{U}}{dt} = \mathbf{L}(\mathbf{U})$$

关键词

Euler equations; Adaptive order; WCNS-AO scheme; Nonlinear interpolation; Shock-capturing capability.

Article ID: A WCNS-AO scheme based on an interpolation method of adaptive order. HU Yinggang, JIANG Yanqun, HUANG Xiaoqian. School of Mathematics and Physics, Southwest University of Science and Technology.

621010 Mianyang

China

Abstract

The system of Euler equations is typically formulated as the form of nonlinear hyperbolic conser- vation law equations. The solutions of this type of equations may have discontinuities which may lead to the emergence of non-physical oscillations near the discontinuities using high-order numerical schemes for solving Euler equations. This paper designs a new weighted compact nonlinear scheme based on an inter- polation method of adaptive order WCNS-AO to solve Euler equations. Following the classical five-point WCNS scheme this scheme uses a nonlinear weighted interpolation technique that combines the low-order interpolations on the substencils and the high-order interpolation on the global stencil to improve the per- formance of the classical scheme. The WCNS-AO scheme combines the sixth-order hybrid cell-edge and cell-node center differencing for flux derivatives at cell-nodes and the nonlinear interpolation technique of adaptive order at cell-edges. The high-order TVD Runge-Kutta method is applied to solve the semi-discrete

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equations generated by spatial discretization of Euler equations. Numerical results show that the WCNS-AO scheme can achieve fifth-order accuracy in smooth domains and has better resolution and shock-capturing capability compared with the classical fifth-order WCNS-Z and WENO-AO schemes.

The Euler equations play a critical role in fluid mechanics. Their two-dimensional form is described as follows:
$$\frac{\partial U}{\partial t} + \frac{\partial F}{\partial x} + \frac{\partial G}{\partial y} = 0$$
where $U$ represents the conservative variables, $F$ and $G$ are the inviscid fluxes, $\rho$ is the fluid density, $u$ and $v$ are the velocity components in the $x$ and $y$ directions, respectively, $E$ is the total energy per unit volume, and $p$ is the pressure. Solutions to the Euler equations may involve phenomena such as shock waves and contact discontinuities, which present significant challenges for both exact analytical solutions and numerical simulations.

Among numerous high-order schemes, the weighted essentially non-oscillatory (WENO) scheme has become an ideal choice for solving the Euler equations. It maintains essentially non-oscillatory properties near discontinuities while achieving high-order accuracy in smooth regions.

Compared to the essentially non-oscillatory (ENO) scheme, which reconstructs numerical fluxes using information from an optimal stencil, the WENO scheme introduces nonlinear weighting techniques. By utilizing information from all candidate stencils to reconstruct the numerical flux, it achieves higher accuracy in smooth regions and maintains stability near discontinuities. Jiang and Shu constructed sub-stencil smoothness indicators and proposed the WENO-JS scheme. To address the issue of accuracy degradation near critical points in the WENO-JS scheme, several improvements have been proposed in the literature. For instance, Henrick introduced a mapping weight function to correct the nonlinear weights, resulting in the WENO-M scheme. While this scheme achieves optimal accuracy near critical points, the use of the mapping weight function significantly increases computational overhead.

Borges defined a global smoothness indicator to design a new type of weight, leading to the WENO-Z scheme. This scheme maintains optimal accuracy near critical points with computational efficiency comparable to WENO-JS. Xu et al. designed a conservative hybrid central-WENO scheme by applying a weighted average of the WENO scheme and a second-order central scheme. Luo et al. further enhanced the resolution of the scheme by adjusting the weights on less-smooth sub-stencils.

Based on a strategy of adaptively selecting between central or upwind stencils, the adaptive central-upwind WENO-CU6 scheme was proposed. This scheme utilizes a central stencil in smooth regions to obtain low-dissipation characteristics and switches to an upwind stencil in discontinuous regions to maintain numerical stability.

Huang et al. improved the WENO-CU6 scheme by using three three-point stencils for reconstruction. In this version, the linear weights can be chosen flexibly (as any positive numbers summing to unity), and a new smoothness indicator was also introduced.

Zhu and Qiu used two-point stencils to construct the WENO-ZQ scheme and the multi-resolution WENO-ZS scheme with nested stencils. Balsara et al. adopted the interpolation concept of the WENO-ZQ scheme to design the WENO-AO scheme, in which the smoothness indicators are formulated as a complete sum of squares.

The weighted compact nonlinear scheme (WCNS) is another high-order, high-resolution method for solving the Euler equations. Deng et al. introduced the concept of nonlinear interpolation into compact nonlinear schemes to propose the WCNS scheme, which possesses high resolution and robust discontinuity-capturing capabilities.

Zhang and Nonomura extended the WCNS scheme to higher orders and demonstrated its superior performance through numerical experiments.

The WCNS scheme not only inherits the advantages of the WENO scheme but also offers three primary benefits compared to classical WENO schemes of the same order: higher resolution, suitability for various flux-splitting forms, and excellent free-stream and vortex preservation on curvilinear grids. Nonomura et al. constructed a WCNS scheme using a hybrid of nodal and cell-edge points, verifying that this scheme is more compact than classical WENO schemes of the same order and provides higher resolution in discontinuous regions.

Based on approximate dispersion relations, it has been verified that the WCNS scheme possesses superior spectral characteristics. Deng et al. developed a local dissipative weighted compact scheme (WCNS-LD). Following the adaptive upwind concept of the WENO-CU scheme, they constructed the WCNS-CU6 and EWCNS-CU4 schemes with adaptive central-upwind properties. These schemes utilize Z-type nonlinear weights to improve shock-capturing capabilities.

Subramaniam et al. improved the WCNS scheme to design a high-order scheme with low dispersion. Hieji and others designed the WCNS-T scheme, which is capable of effectively capturing high-frequency waves and strong shocks. Fan et al. designed a WCNS scheme for solving shallow water equations with source terms. Jiang et al. proposed a high-order semi-implicit WCNS scheme, which circumvents strict stability limits and improves computational efficiency.

Drawing on the WENO-AO interpolation strategy from the literature, this study constructs a WCNS-AO scheme that combines low-order interpolation on hybrid three-point sub-stencils with high-order interpolation on a global five-point stencil for solving the Euler equations. The WCNS-AO spatial discretization scheme integrates a hybrid nodal/cell-edge sixth-order central difference scheme with an adaptive-order nonlinear weighted interpolation method based on characteristic projection. To verify the performance of the WCNS-AO scheme, it is coupled with a high-order TVD Runge-Kutta time integration method and applied to one-dimensional and two-dimensional numerical cases. The results are compared with the classical fifth-order WCNS-Z and fifth-order WENO-AO schemes.

1 数值方法

1. Numerical Method Derivation

In this section, we present the derivation of the fifth-order Weighted Compact Nonlinear Scheme (WCNS) using the one-dimensional Euler equations as a representative example. For higher-dimensional cases, the derivation can be extended analogously using a dimension-by-dimension approach. The one-dimensional Euler equations are described as follows:

$$\frac{\partial U}{\partial t} + \frac{\partial F(U)}{\partial x} = 0$$

We employ a uniform grid where $\Delta x$ represents the grid spacing and $N$ denotes the number of grid points. The semi-discrete form of the governing equations is given by:

$$\frac{dU_i}{dt} = -\left( \frac{\partial F}{\partial x} \right)_i$$

1.1 Sixth-Order Central Difference Scheme

Based on the classical fifth-order WCNS framework, the flux derivative $\left( \frac{\partial F}{\partial x} \right)_i$ is computed using a hybrid central difference scheme involving both node and half-node points:

$$\left( \frac{\partial F}{\partial x} \right)i = \frac{1}{\Delta x} \sum$$} a_j \tilde{F}_{i+j

[FIGURE:1]

As illustrated in [FIGURE:1], the flux values at the half-nodes, $F_{i+1/2}$, are not directly available and must be obtained through an interpolation process. To ensure numerical stability and capture discontinuities without oscillations, these half-node fluxes are typically reconstructed using a weighted combination of candidate stencils.

1.2 Fifth-Order WCNS Interpolation

The core of the WCNS-Z scheme lies in the high-order interpolation of the numerical flux at the cell interfaces. For the fifth-order scheme, the values at the half-node $i+1/2$ are determined by considering a five-point stencil. This stencil is subdivided into three smaller sub-stencils, and a weighted average of the interpolations from these sub-stencils is used to achieve both high-order accuracy in smooth regions and shock-capturing capabilities near discontinuities.

[TABLE:1]

The specific weights are calculated based on local smoothness indicators, which allow the scheme to automatically switch between a high-order central-like interpolation and a lower-order upwind-biased interpolation depending on the solution's behavior. This approach effectively suppresses Gibbs oscillations near shocks while maintaining sixth-order formal accuracy in smooth regions of the flow.

In the equation, the scheme is referred to as an implicit scheme when the parameter $\alpha \neq 0$; otherwise, it is known as an explicit scheme. By performing a Taylor expansion on the equation, the relationship between the order of accuracy and the parameter $\alpha$ can be derived. Under normal circumstances, the choice between implicit and explicit schemes does not significantly impact the numerical results. However, explicit schemes are computationally more efficient than implicit schemes because they do not require solving a tridiagonal linear system to obtain the flux derivative approximations. Consequently, the present study employs the following sixth-order explicit scheme to calculate the flux derivatives.

$$F'i = \frac{64}{45h} (\tilde{F}} - \tilde{F{i-1/2}) - \frac{4}{45h} (\tilde{F}$$} - \tilde{F}_{i-3/2}) \tag{5

In the formula, $\tilde{F}$ represents the numerical approximation of the flux at the half-node, which is obtained through a nonlinear weighted interpolation method processed by characteristic projection. The Jacobian matrix of the characteristic projection method represents the speed of sound. Performing the eigenvalue decomposition of the Jacobian matrix yields $\Lambda = \text{diag}(\dots)$

$$A = \frac{\partial F}{\partial U} = R \Lambda L$$

$R$ is the characteristic matrix, where $L$ and $R$ are the left and right eigenvector matrices, respectively. Let $\lambda_{max} = \max { |u + c|, |u - c| }$ denote the maximum characteristic speed of the Jacobian matrix.

To improve the stability of the numerical scheme, the Lax-Friedrichs splitting technique is employed to decompose the flux at each node into positive and negative components.

$$F^{\pm}_i = (F_i \pm \lambda U_i) / 2$$

the maximum eigenvalue (spectral radius) of the Jacobian matrix. Below, we provide only the calculation process for the left state; the calculation for the right state at the half-node is mirror-symmetric and follows a similar procedure, and is thus omitted for brevity. To ensure a concise derivation, the superscript notation is also omitted. The formula for calculating the numerical flux at the nodes is as follows:

[TABLE:1]

To effectively eliminate non-physical oscillations near discontinuities, this study employs non-linear weighted interpolation processed via characteristic projection. First, the flux $\mathbf{F}$ at each node is projected into the characteristic space to obtain the characteristic flux components.

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In the characteristic space, a nonlinear weighted interpolation method is employed to solve for the numerical characteristic fluxes. Finally, these fluxes are projected back into the original space to obtain the physical flux. The left and right characteristic vector matrices are determined using the arithmetic mean of the values from two adjacent nodes.

$$\mathbf{F}{i+1/2} = R$$} \tilde{\mathbf{W}}_{i+1/2

3 WCNS-AO

To obtain the fifth-order accurate WCNS-AO scheme, a five-point stencil is employed to calculate the numerical characteristic fluxes at the half-node locations. The following describes the nonlinear weighted interpolation process for one of the components; other components are processed analogously. By utilizing the Lagrange interpolation polynomial function $P(x)$, the fifth-order numerical flux is calculated as follows:

$$ \hat{f}{i+1/2} = \sum $$}^{4} \omega_j f_{i-2+j

The above expression can also be derived by taking the linear combination of third-order numerical fluxes calculated on three sub-stencils. Here, the sub-stencils are defined as:

[FIGURE:1]

Specifically, the fifth-order numerical flux is obtained through the linear weighting of the third-order fluxes from these sub-stencils. This approach ensures that the scheme maintains high-order accuracy in smooth regions while providing necessary stability near discontinuities.

The Lagrange interpolating polynomial function is used to calculate the third-order numerical flux. Specifically, the high-order numerical flux can be expressed as a linear combination of low-order numerical fluxes, given by:
$$\tilde{w}{i+1/2} = \sum$$}^{2} \gamma_k \tilde{w}_{k, i+1/2
In this expression, the linear weights are denoted as $\gamma$. To improve the performance of the original fifth-order WCNS-Z scheme in terms of resolution and shock-capturing capabilities, this study adopts a non-linear weighted interpolation technique based on the WENO-AO (Adaptive Order) interpolation framework. This approach utilizes a hybrid of low-order interpolation on sub-stencils and high-order interpolation on the global stencil to calculate the values at the cell interface $i+1/2$.

By substituting the numerical flux $\tilde{w}$ from Eq. (9), the computational scheme is rewritten as:

where $\epsilon$ is an arbitrary positive constant that must satisfy:

$k = 0$ and $k = 1$. In this study,

While high-order schemes can be obtained using linear weights, numerical oscillations tend to occur near discontinuities. To address this issue, the concept of nonlinear interpolation is employed, where nonlinear weights replace linear weights. This approach ensures high-order accuracy in smooth regions while avoiding interpolation across discontinuities. Various definitions of nonlinear weights have been proposed in the literature \cite{}, among which the $Z$-type weights (WENO-Z) are particularly popular. To derive the nonlinear weighted form of Eq. (16), we first define the smoothness indicators for each candidate stencil.

The scheme utilizes an interpolative numerical flux. The smoothness indicators for each stencil can be defined by the sum of the squares of the derivatives of the interpolation polynomials. Specifically, the smoothness indicator for each stencil is given by:

$$\beta_3 = (h p')^2 + (h^2 p'')^2 + (h^3 p^{(3)})^2 + (h^4 p^{(4)})^2 \tag{13}$$

The specific expression is:

The global smoothness indicator is adopted from the method proposed in \cite{Reference_Placeholder}. By modifying the linear weights into non-linear weights and combining them with the corresponding equations, the WCNS-AO scheme is obtained. The non-linear interpolation format for the numerical flux $\hat{f}$ is derived by performing a Taylor expansion of each smoothness indicator at the point $x_j$, yielding:

When this occurs, we can obtain

$\tau = O(h^4) \tag{18}$. When $w'_i = 0$, we obtain:

$\tau = O(h^4) \tag{19}$ Substituting (18) and (19) into (15), we obtain

$\omega_k = d_k + O(h^4) \tag{20}$ Rewriting the nonlinear interpolation scheme (16) as

$$ \hat{f}{i+1/2} = \tilde{w} + O(h^5) + O(h^6) $$

is a constant. Combined with Eq. (21), it can be proven that the nonlinear interpolation scheme for the numerical flux achieves fifth-order accuracy. After applying the WCNS-AO scheme for spatial discretization of the Euler equations, we obtain a system of ordinary differential equations (ODEs) with respect to time $t$:

$$ \begin{aligned} U^{(1)} &= U^n + \Delta t L(U^n) \ U^{(2)} &= \frac{3}{4} U^n + \frac{1}{4} U^{(1)} + \frac{1}{4} \Delta t L(U^{(1)}) \ U^{(n+1)} &= \frac{1}{3} U^n + \frac{2}{3} U^{(2)} + \frac{2}{3} \Delta t L(U^{(2)}) \end{aligned} $$

Introduction

In the context of time-series analysis and dynamic system modeling, the temporal dimension is discretized into specific intervals. We define $\Delta t$ as the time step, which represents the fundamental unit of temporal resolution in our simulations and numerical calculations.

2 数值实验

This section evaluates the performance of the fifth-order WCNS-AO scheme through several numerical experiments and compares it with the fifth-order WENO-AO scheme. The accuracy test case utilizes the Euler equations with the initial conditions set as:

$\rho(x, 0) = 2 + \sin[x - \sin(x)]$, $u = 1$, and $P = 1$. The exact solution is given by $\rho(x, t) = 2 + \sin[(x - t) - \sin(x - t)]$.

The computational domain is $[0, 2\pi]$, and the final simulation time is $t = 2.0$ s. Periodic boundary conditions are applied. To eliminate the influence of time discretization on the spatial order of accuracy, a sufficiently small time step is used. Based on the numerical errors and convergence orders of the WCNS-AO and WENO-AO schemes, it can be observed that for this test case, the WCNS-AO scheme exhibits better convergence properties than the WENO-AO scheme and successfully achieves fifth-order accuracy in smooth regions.

Based on the numerical errors and accuracy orders observed in the one-dimensional problems, and...

1 Numerical errors

convergence rates and CPU time for the 1D case

方法

WCNS-AO 2. 32 × 10 6. 62 × 10 1. 14 × 10 4. 21 × 10 1. 60 × 10 6. 12 × 10 3. 86 × 10 1. 48 × 10 1. 28 × 10 4. 91 × 10 WENO-AO 2. 23 × 10 6. 66 × 10 1. 21 × 10 4. 22 × 10 1. 78 × 10 6. 12 × 10 4. 53 × 10 1. 48 × 10 1. 60 × 10 5. 91 × 10

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Consider the two-dimensional Euler equations with the initial condition set as $\rho(x, y, 0) = 1 + 0.2 \sin(x + y)$, $u(x, y, 0) = 1$, $v(x, y, 0) = 1$, and $p(x, y, 0) = 1$. The computational domain is $[0, 2\pi] \times [0, 2\pi]$ with periodic boundary conditions, and the simulation is conducted until the final time $T = 2.0$ s. [TABLE:1] presents the numerical errors, convergence orders, and CPU run times for the WCNS-AO and WENO-AO schemes.

As shown in [TABLE:1], both schemes achieve their theoretical fifth-order accuracy in smooth regions. Furthermore, [FIGURE:1] illustrates the relationship between numerical error and computational time for these two schemes. The results indicate that while both methods maintain high precision, the WCNS-AO scheme demonstrates competitive efficiency in terms of balancing error reduction and computational overhead.

2 Lax

This problem is a classic one-dimensional Riemann problem, where the initial values contain both rarefaction waves and shock waves. It is commonly used to test the shock-capturing capabilities of numerical schemes. The initial conditions are $\mathcal{U}_0$, and the computational domain is $[0, 1]$ with compactly supported boundary conditions.

The number of grid points is $N = 200$, and the computation termination time is $T = 0.13$ s. [FIGURE:1] presents the numerical results for the fifth-order WENO-AO and WCNS-AO schemes, compared against a reference solution obtained from a fine grid ($N = 2000$).

Problem: Comparison of numerical solutions of different schemes. The numerical error of the WCNS-AO scheme is lower than that of the WENO-AO scheme. [FIGURE:2] shows the error as a function of time.

As shown in the figure, the computational efficiencies of the two schemes are comparable for one-dimensional problems. However, for two-dimensional problems, the computational efficiency of WCNS-AO is slightly higher than that of WENO-AO.

Numerical errors, orders of accuracy, and computational efficiency based on two-dimensional problems are presented in [TABLE:1].

2 Numerical errors

convergence rates and CPU time for the 2D case

方法

WCNS-AO 40 × 40 6. 02 × 10 9. 47 × 10 50 × 50 1. 98 × 10 3. 11 × 10 60 × 60 7. 96 × 10 1. 25 × 10 70 × 70 3. 69 × 10 5. 79 × 10 80 × 80 1. 89 × 10 2. 97 × 10 WENO-AO 40 × 40 4. 16 × 10 6. 98 × 10 50 × 50 1. 36 × 10 2. 36 × 10 60 × 60 5. 49 × 10 9. 78 × 10 70 × 70 2. 54 × 10 4. 56 × 10 80 × 80 1. 30 × 10 2. 34 × 10

As shown in the figure, the WENO-AO scheme produces significant numerical oscillations near the shock wave. In contrast, the WCNS-AO scheme exhibits the best performance, effectively suppressing these oscillations and demonstrating superior shock-capturing capabilities.

3 Sod

This problem is a classic one-dimensional Riemann problem, commonly used to test the stability of numerical schemes. The initial conditions are given by (25). The computational domain is defined as $[0, 1]$, and compact support boundary conditions are applied. The number of grid points is set to $N = 400$, and the simulation is terminated at $t = 0.13$ s. [FIGURE:1] presents the numerical results for the WENO-AO and WCNS-AO schemes compared against the reference solution. The numerical results demonstrate that both schemes exhibit excellent stability. Among them, the WCNS-AO scheme shows the best performance in shock capturing and superior overall stability.

Problem: Comparison of numerical solutions of different schemes

4 Osher-Shu

The solution to this problem exhibits high-frequency oscillatory characteristics and is frequently employed to evaluate the accuracy and dissipation errors of numerical schemes. The initial conditions are defined as:

In Equation (26), the computational domain is defined as $[-0.5, 0.5]$. Compactly supported boundary conditions are implemented at the boundaries of this domain.

[FIGURE:1]

To verify the accuracy and convergence of the proposed numerical method, we conduct a series of simulations across varying grid resolutions. The error analysis is performed by comparing the numerical results with the analytical solution provided in $\cite{ref1}$. As shown in [TABLE:1], the method demonstrates second-order convergence in the $L_2$ norm, consistent with the theoretical expectations for the spatial discretization scheme employed.

[TABLE:1]

Furthermore, the stability of the solution is maintained even when the CFL number approaches the theoretical limit. This robustness is particularly evident in the treatment of the source terms, where the balanced formulation prevents the development of spurious oscillations at the interface. The results for the primary variables, including the velocity field and pressure distribution, are plotted in [FIGURE:2], illustrating the high fidelity of the simulation in capturing sharp gradients.

The computational grid consists of $201 \times 201$ points, and the simulation is terminated at $t = 1.8$ s. The numerical results obtained from various schemes are compared against the reference solution. These results demonstrate that, compared to the standard WENO scheme of the same order, the WCNS-AO scheme performs significantly better at the interface between acoustic and entropy waves. Furthermore, the WCNS-AO scheme exhibits a superior ability to capture high-frequency waves, providing higher overall resolution.

Comparison of Numerical Solutions for the Riemann Problem

The Riemann problem contains a wide variety of flow structures, making it an ideal benchmark for evaluating the stability, dissipation, resolution, and shock-capturing capabilities of the constructed numerical schemes. Within the computational domain, we consider the initial conditions as follows:

[TABLE:1]

For this simulation, Dirichlet boundary conditions are applied with a grid resolution of $401 \times 401$. The computation is terminated at time $t = 0.8$ s.

The density distribution results for the various schemes are presented, featuring density contours ranging from the minimum to maximum values. As illustrated in the figures, the proposed schemes are capable of capturing the majority of the vortex structures. Notably, the WCNS-AO scheme demonstrates superior performance compared to the WENO-AO scheme. It is capable of resolving much finer vortex structures, indicating lower numerical dissipation and higher spatial resolution.

[FIGURE:1]

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Comparison of Numerical Solutions for the Riemann Problem

This problem describes a complex flow phenomenon involving shock wave reflection and vortex structures, caused by an oblique shock wave striking an inviscid wall at a specific angle. This test case is commonly used to evaluate the ability of numerical schemes to capture both shock waves and small-scale structures. The computational domain is defined as $[0, 4] \times [0, 1]$, with the initial conditions given as follows:

$$
\begin{aligned}
(\rho, u, v, p) =
\begin{cases}
(1.0, 0.0, 0.0, 1.0), & \text{if } y > (x - 0.16) \tan(60^\circ) \
(1.691, 2.319, 0.0, 10.0), & \text{otherwise}
\end{cases}
\end{cases}
$$

The boundary conditions are prescribed as follows: the left boundary and the bottom segment where $x < 0.16$ are set as inflow boundaries; the remaining portion of the bottom boundary ($x \ge 0.16$) is set as a reflective wall; the right boundary is an outflow boundary; and the top boundary is set according to the exact solution. The simulation is performed on a grid of $400 \times 100$ cells, with a final computation time of $t = 0.25$.

[FIGURE:1]

[TABLE:1]

The density distribution results obtained from the different schemes are presented in [FIGURE:1], with density contours ranging from 1.0 to 1.7. While all schemes are capable of capturing the primary shock wave information within the flow field, there are notable differences in their performance regarding fine-scale features. Compared to the WENO-AO scheme, the WCNS-AO scheme captures more detailed vortex structures in the roll-up region. This demonstrates that the WCNS-AO scheme possesses lower numerical dissipation and superior resolution for resolving complex flow patterns.

7 Rayleigh-Taylor

Richtmyer-Meshkov Instability

The Richtmyer-Meshkov instability (RMI) is a phenomenon triggered when an interface between two fluids of different densities is impulsive accelerated, typically by a shock wave. In this study, the computational domain is defined as $[0, 0.25] \times [0, 1.0]$. The initial conditions are specified as follows:

[TABLE:1]

For this numerical simulation, the ratio of specific heats is set to $\gamma = 1.4$. Reflective boundary conditions are applied to the left and right boundaries, while the top boundary is set to $(\rho, u, v, p) = (1.0, 0, 0, 1.0)$ and the bottom boundary is set to $(\rho, u, v, p) = (0.125, 0, 0, 0.1)$.

The computational grid consists of $241 \times 961$ cells, and the simulation is executed until the termination time $t = 0.02$.

[FIGURE:1] presents the density distribution results obtained using different numerical schemes. As illustrated in the figure, the WCNS-AO scheme demonstrates superior performance compared to the WENO-AO scheme. Specifically, WCNS-AO is capable of capturing the fine-scale vortices induced by the Richtmyer-Meshkov instability at the density interface with higher fidelity.

In this case, the results produced by the WCNS-AO scheme are slightly better, exhibiting the lowest numerical dissipation among the tested methods. A detailed comparison of the numerical solutions for the instability problem across different schemes is provided below.

3 结

In this study, an adaptive-order weighted compact nonlinear scheme (WCNS-AO) is constructed for the Euler equations. This scheme combines a hybrid node and semi-node sixth-order central difference format for calculating the spatial derivatives of flux at the computational cell nodes with an adaptive-order nonlinear interpolation method for calculating the numerical flux at the semi-nodes.

The WCNS-AO scheme employs flux splitting techniques and characteristic projections to enhance stability and effectively eliminate non-physical oscillations near discontinuities. Combined with a third-order Runge-Kutta time discretization method, the newly designed fifth-order WCNS-AO scheme, the classic fifth-order WCNS scheme, and the fifth-order WENO-JS scheme were respectively applied to solve one-dimensional and two-dimensional Euler equations. Numerical results demonstrate that the WCNS-AO scheme achieves the designed fifth-order accuracy in smooth regions, and its computational efficiency is slightly higher than that of the WENO-AO scheme.

Numerical examples, such as the Shock-Tube problem, verify that the WCNS-AO scheme exhibits superior shock-capturing capabilities and better stability compared to the WENO-AO scheme of the same order. Results from the two-dimensional Riemann problem and the Double Mach Reflection problem further confirm that the WCNS-AO scheme outperforms the WENO-AO scheme, as it is capable of capturing finer flow field structures with lower numerical dissipation and higher resolution.

Regarding stability issues, while the WENO-AO scheme performs slightly better than the standard WCNS scheme in certain cases, the WCNS-AO scheme demonstrates superior overall performance, thereby improving the dissipation characteristics of the original format.

References

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